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That’s a very good question Tom.
The common self adjusting 6 jaw used in horology is often referred to as a bezel chuck. As the name implies it’s mostly used to hold bezels to re-groove or to do basic machining of bezels and case backs. These chucks are also good when turning parts with thin walls for both internal and external clamping of a workpiece. Having double the jaws of a three jaw and two more jaws than a 4 jaw the amount of pressure required to hold a part is distributed to these additional jaws thus reducing clamping pressure at any given clamping point which can be helpful when machining any part that may be sensitive to clamping pressure.
Back to your question about what can be accomplished using a 6 jaw over a 4 jaw. Modifications such as soft jaws bored or turned to necessary diameters can be applied to three and 4 jaw chucks to help distribute clamping pressure and protect surfaces etc. which would effectively do as much or possibly more than a 6 jaw.
As David has mentioned before, the 4 jaw is capable of far more accuracy than a self adjusting 3 jaw which is also true when compared to the self adjusting 6 jaw bezel chuck. There are independent 6 jaw chucks available on the market, some with micro adjustment capabilities which should give you the accuracy of a 4 jaw but my small bezel chucks for horological use are adjusted by turning a knurled outer ring. Other than shimming for slightly eccentric parts there’s no real built in independent adjusting feature in my bezel chucks. I do use a 3 jaw far more often than a 4 jaw and tend to use collets or turn between centers when close tolerances are needed. Since watch part diameters usually fall well within available collet sizes I would guess that I turn at least 50% of the time using collets, step and wax chucks, face plate, expanding plugs etc. 30% turning and truing between centers, 15% of the time with a 3 jaw, and maybe 5% of the time with a 4 jaw and bezel chuck. Of course these figures are a guesstimate and apply to watches. These usage percentages (for me) would lean much more towards the 3 and 4 jaw chucks when dealing with larger clocks unless equipped with a larger lathe using 3,4 or 5c collets. I added my preferences here just to give you an idea of what works for me. Others approach the same jobs differently with equally good results. David as you know prefers using the 4 jaw any day over a 3 jaw . I can’t argue with his reasoning for this preference as he supports it with perfect logic. The 4 jaw can be much more accurate and requires very little time for him to true the part. Soooo…. if you follow my approach then the 4 and 6 jaw usage may be fairly even with the 4 jaw usage a bit ahead of the 6 with the 6 jaw having the advantage of distributing clamping pressure out to 2 more jaws with more clamping points to help eliminate flexing which can occur when working with a thin, large diameter bezel. If you lean more towards Davids approach then I would feel the 4 jaw would prove far more valuable and useful to you than a 6 jaw except for the reasons I mention above.
I try not to answer questions that were not even asked but I sometimes get carried away and give more info than was asked for or is necessary (as I’ve done here) . I added a bit extra just to give you some different perspectives. I hope it’s helpful Tom,